Practice the questions given in the worksheet on Remainder Theorem.

1. Use the Remainder Theorem, find the remainder when 4x\(^{3}\) - 3x\(^{2}\) + 2x - 4 is divided by x + 1.

2. If p(y) = y\(^{3}\) + y\(^{2}\) - 2y + 1, using Remainder Theorem, find the remainder, when p(y) is divided by (y – 3), find the value of p(a).

3. Find the remainder (without division) when

(a) x\(^{2}\) - 2x + 4 is divided by x - 1

(b) 2x\(^{3}\) - 3x\(^{2}\) + 7x - 8 is divided by x - 1

4. Use the Remainder Theorem, find the remainder when x\(^{4}\) - 3x\(^{2}\) + 4x - 12 is divided by x - 3.

5. Find the remainder (without division) when

(a) x\(^{3}\) + 4x + 2 is divisible by x + 2

(b) 4x\(^{3}\) - 3x\(^{2}\) + 5x + 4 is divided by 2x + 1

(c) 4x\(^{3}\) + 5x\(^{2}\) + 6x - 7 is divided by 2x - 1

6. What number should be added to x\(^{2}\) + 5 so that the resulting polynomial leaves the remainder 3 when divided by x + 3?

7. Use the Remainder Theorem, find the remainder when 4x\(^{3}\) - 3x\(^{2}\) + 2x - 4 is divided by x + 1.

8. What number should be subtracted from 3x\(^{2}\) + 5x so that the resulting polynomial leaves the remainder 1 when divided by 2x + 5?

9. Use the Remainder Theorem, find the remainder when x\(^{6}\) + 3x\(^{2}\) + 10 is divided by x - 2.

10. Find a if the remainder is a when x\(^{3}\) + 3x\(^{2}\) - ax + 3 is divided by x - 2.

11. If the polynomials ax\(^{3}\) + 4x\(^{2}\) + 3x – 4 and x\(^{3}\) - 4x + a leave the same remainder when divided by (x - 3), find the value of a.

12. Find the value of k if the remainder is -3 when kx\(^{3}\) + 8x\(^{2}\) - 4x + 10 is divided by x +1.

13. If both ax\(^{3}\) + 2x\(^{2}\) - 3 and x\(^{2}\) - ax + 4 leave the same remainder when divided by x - 2, find a.

Answers for the worksheet on Remainder Theorem are given below:

**Answers:**

1. -13

2. 31, a\(^{3}\) + a\(^{2}\) - 2a + 1

3. (a) 3

(b) -2

4. 54

5. (a) -14

(b) \(\frac{1}{4}\)

(c) -\(\frac{9}{4}\)

6. -11

7. -13

8. \(\frac{21}{4}\)

9. 86

10. \(\frac{23}{3}\)

11. a = -1.

12. 25

13. \(\frac{3}{10}\)

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